Speeding up the constraint-based method in difference logic

"The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-319-40970-2_18"Over the years the constraint-based method has been successfully applied to a wide range of problems in program analysis, from invariant generation to termination and non-termination proving. Quite often the semantics of the program under study as well as the properties to be generated belong to difference logic, i.e., the fragment of linear arithmetic where atoms are inequalities of the form u v = k. However, so far constraint-based techniques have not exploited this fact: in general, Farkas’ Lemma is used to produce the constraints over template unknowns, which leads to non-linear SMT problems. Based on classical results of graph theory, in this paper we propose new encodings for generating these constraints when program semantics and templates belong to difference logic. Thanks to this approach, instead of a heavyweight non-linear arithmetic solver, a much cheaper SMT solver for difference logic or linear integer arithmetic can be employed for solving the resulting constraints. We present encouraging experimental results that show the high impact of the proposed techniques on the performance of the VeryMax verification systemPeer ReviewedPostprint (author's final draft

On Optimization Modulo Theories, MaxSMT and Sorting Networks

Optimization Modulo Theories (OMT) is an extension of SMT which allows for finding models that optimize given objectives. (Partial weighted) MaxSMT --or equivalently OMT with Pseudo-Boolean objective functions, OMT+PB-- is a very-relevant strict subcase of OMT. We classify existing approaches for MaxSMT or OMT+PB in two groups: MaxSAT-based approaches exploit the efficiency of state-of-the-art MAXSAT solvers, but they are specific-purpose and not always applicable; OMT-based approaches are general-purpose, but they suffer from intrinsic inefficiencies on MaxSMT/OMT+PB problems. We identify a major source of such inefficiencies, and we address it by enhancing OMT by means of bidirectional sorting networks. We implemented this idea on top of the OptiMathSAT OMT solver. We run an extensive empirical evaluation on a variety of problems, comparing MaxSAT-based and OMT-based techniques, with and without sorting networks, implemented on top of OptiMathSAT and {\nu}Z. The results support the effectiveness of this idea, and provide interesting insights about the different approaches.Comment: 17 pages, submitted at Tacas 1

Dumps for dead livestock and the conservation of wintering Red Kites (Milvus milvus)

Volume: 33Start Page: 338End Page: 34

On Solving Universally Quantified Horn Clauses

Abstract. Program proving can be viewed as solving for unknown relations (such as loop invariants, procedure summaries and so on) that occur in the logical verification conditions of a program, such that the verification conditions are valid. Generic logical tools exist that can solve such problems modulo certain background theories, and therefore can be used for program analysis. Here, we extend these techniques to solve for quantified relations. This makes it possible to guide the solver by constraining the form of the proof, allowing it to converge when it otherwise would not. We show how to simulate existing abstract domains in this way, without having to directly implement program analyses or make certain heuristic choices, such as the terms and predicates that form the parameters of the abstract domain. Moreover, the approach gives the flexibility to go beyond these domains and experiment quickly with various invariant forms.

Minimal-model-guided approaches to solving polynomial constraints and extensions

In this paper we present new methods for deciding the satisfiability of formulas involving integer polynomial constraints. In previous work we proposed to solve SMT(NIA) problems by reducing them to SMT(LIA): non-linear monomials are linearized by abstracting them with fresh variables and by performing case splitting on integer variables with finite domain. When variables do not have finite domains, artificial ones can be introduced by imposing a lower and an upper bound, and made iteratively larger until a solution is found (or the procedure times out). For the approach to be practical, unsatisfiable cores are used to guide which domains have to be relaxed (i.e., enlarged) from one iteration to the following one. However, it is not clear then how large they have to be made, which is critical. Here we propose to guide the domain relaxation step by analyzing minimal models produced by the SMT(LIA) solver. Namely, we consider two different cost functions: the number of violated artificial domain bounds, and the distance with respect to the artificial domains. We compare these approaches with other techniques on benchmarks coming from constraint-based program analysis and show the potential of the method. Finally, we describe how one of these minimal-model-guided techniques can be smoothly adapted to deal with the extension Max-SMT of SMT(NIA) and then applied to program termination proving.Peer Reviewe